Using synergistic shapely values to increase robustness of a cellular network

ABSTRACT

A method and associated systems for improving robustness of a cellular network. A topology of the cellular network is represented as an undirected graph that represents base stations as nodes and represents communication paths between base stations as edges. Each node is associated with a “synergistic” version of a Shapley value proportional to an amount of network disturbance that would occur if that value&#39;s corresponding base station should fail. The synergistic nature of the synergistic Shapley values allows them to account for scenarios in which multiple base stations fail at the same time. A synergistic Shapley value of a particular node is derived as a function of how many of the shortest paths between nodes of the graph lengthen when the node&#39;s corresponding base station fails. Base stations and nodes associated with higher synergistic Shapley values are deemed to be “censorious” and in need of reconfiguration.

This application is a continuation application claiming priority to Ser.No. 15/675,905, filed Aug. 14, 2017, now U.S. Pat. No. 10,057,791 issuedAug. 21, 2018, which is a continuation application of Ser. No.15/188,529, filed Jun. 21, 2016, U.S. Pat. No. 9,794,805, issued Oct.17, 2017.

TECHNICAL FIELD

The present invention relates to improving reliability of cellularnetworks by mitigating adverse effects a technical problem that disablesa base station.

BACKGROUND

In a cellular network, data is transmitted between cellular basestations distributed throughout geographical region covered by thenetwork. A base station failure may force data to be rerouted through alengthier, less-efficient, or more congested path. Because a largecellular network may comprise very large numbers of base stations, onetechnical problem inherent in cellular technology is the difficulty inorganizing base stations so as to minimize the impact of a failure. Inparticular, the complex topology of a typical cellular network makes ithard to identify which base station failures would be must disruptive.This problem becomes far more challenging when attempting to considerthe possibility of multiple concurrent base-station failures.

BRIEF SUMMARY

A first embodiment of the present invention providesnetwork-optimization system comprising a processor, a memory coupled tothe processor, and a computer-readable hardware storage device coupledto the processor, the storage device containing program code configuredto be run by the processor via the memory to implement a method forimproving robustness of a cellular network, the method comprising:

receiving an undirected graph that represents a topology of a cellularnetwork, where the graph represents a pair of base stations of thenetwork as a pair of nodes, and where the graph represents acommunications path between the pair of base stations as an edgeconnecting the pair of nodes;

deriving, for each node of the graph, a corresponding Shapley value of aplurality of Shapley values;

identifying a first node of the graph as a censorious node if a Shapleyvalue that corresponds to the first node exceeds a predeterminedthreshold value; and

updating the graph by rearranging one or more censorious nodes of thegraph so as to reduce the Shapley values that correspond to the one ormore censorious nodes.

A second embodiment of the present invention provides method forimproving robustness of a cellular network, the method comprising:

receiving an undirected graph that represents a topology of a cellularnetwork, where the graph represents a pair of base stations of thenetwork as a pair of nodes, and where the graph represents acommunications path between the pair of base stations as an edgeconnecting the pair of nodes;

deriving, for each node of the graph, a corresponding Shapley value of aplurality of Shapley values;

identifying a first node of the graph as a censorious node if a Shapleyvalue that corresponds to the first node exceeds a predeterminedthreshold value; and

updating the graph by rearranging one or more censorious nodes of thegraph so as to reduce the Shapley values that correspond to the one ormore censorious nodes.

A third embodiment of the present invention provides a computer programproduct, comprising a computer-readable hardware storage device having acomputer-readable program code stored therein, the program codeconfigured to be executed by a network-optimization system comprising aprocessor, a memory coupled to the processor, and a computer-readablehardware storage device coupled to the processor, the storage devicecontaining program code configured to be run by the processor via thememory to implement a method for improving robustness of a cellularnetwork, the method comprising:

receiving an undirected graph that represents a topology of a cellularnetwork, where the graph represents a pair of base stations of thenetwork as a pair of nodes, and where the graph represents acommunications path between the pair of base stations as an edgeconnecting the pair of nodes;

deriving, for each node of the graph, a corresponding Shapley value of aplurality of Shapley values;

identifying a first node of the graph as a censorious node if a Shapleyvalue that corresponds to the first node exceeds a predeterminedthreshold value; and

updating the graph by rearranging one or more censorious nodes of thegraph so as to reduce the Shapley values that correspond to the one ormore censorious nodes.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the structure of a computer system and computer programcode that may be used to implement a method for improving robustness ofa cellular network in accordance with embodiments of the presentinvention.

FIG. 2 is an undirected graph, representing an organization of asix-macrocell cellular network, that may be optimized by embodiments ofthe method of the present invention.

FIG. 3 is a flow chart that illustrates the steps of a method forimproving robustness of a cellular network in accordance withembodiments of the present invention.

FIG. 4 shows details of a method for computing synergistic Shapleyvalues of nodes of a graph in accordance with embodiments of the presentinvention.

FIG. 5 is an undirected graph representing an organization of afive-macrocell cellular network that may be optimized by embodiments ofthe method of the present invention.

DETAILED DESCRIPTION

Embodiments of the present invention may be used to optimize any sort ofnetwork or arrangement of connected nodes that may be represented as adirected or undirected graph. Embodiments and examples contained in thisdocument describe, for pedagogical reasons, applications in the field ofcellular-networking technology, but they should not be construed tolimit the scope of the present invention to cellular networks. Otherembodiments may, for example, be straightforwardly applied to computerLANs, cloud-based infrastructure, interpersonal communications networks,social networks, electronic media-distribution channels, supply-chainnetworks, or any other types of entities that may represented as adirected or undirected graph. Furthermore, although examples andembodiments described in this document disclose methods of optimizingorganizations of macrocells in a cellular network, other embodimentsrelated to cellular-network technology might be instead optimizeorganizations of other entities of a cellular network, such asmicrocells or femtocells.

In a cellular network, data is transmitted between base stationsdistributed throughout a geographical region covered by the network. Abase station failure may force data that it would otherwise have managedto be rerouted through a lengthier, less-efficient, or more congestedpath. Because a large cellular network may comprise very large numbersof base stations, one technical problem inherent in cellular technologyis the difficulty in organizing base stations so as to minimize theimpact of a failure.

In particular, the complex topology of a typical cellular network makesit hard to identify which base station failures would be most disruptiveby, for example, forcing network traffic through a longer orless-efficient path or by increasing the lengths of the shortest routesbetween pairs of other base stations. This identification problembecomes far more challenging when attempting to account for thepossibility of multiple concurrent base-station failures.

A macrocell of a cellular network may be the largest logical unit ofradio coverage supported by the network, and may encompass the coveragearea of a cell tower, or “base station.” Each macrocell receives datafrom and sends data to one or more adjacent macrocells, and the overallorganization of macrocells in a cellular network, or in a subset of acellular network, may be represented topologically as an undirectedgraph. In this document, a node of such a graph may equivalentlyrepresent a macrocell, a base station, or a cell tower. FIG. 2 shows anexample of such a graph.

When a base station fails or becomes so congested that its correspondingmacrocell no longer operates satisfactorily, a telecom carrierresponsible for the cellular network may need to quickly identify thefailing base station and reroute traffic around that base station. Apoorly designed network will comprise numerous “censorious” basestations that, when failing, have the greatest negative impact upon thenetwork, thus creating the greatest burdens.

The term “censorious” may be a qualitative term that merely describes abase station or macrocell that, upon failure, creates an unacceptabledegree of negative impact upon the network. Embodiments of the presentinvention, on the other hand, use quantitative measures to moredeterministically identify censorious base stations. Theseimplementation-dependent measures may, for example, comprise identifyinga base station as censorious as a function of whether its failure wouldcause more than a certain number of network paths to be lengthened.

More specifically, embodiments of the present invention identifycensorious entities, such as base stations or macrocells, by assigningeach one an enhanced version of a synergistic Shapley value parameter.synergistic Shapley values are known to those skilled in the art of gametheory as a means of assigning a unique distribution of a total surplusof a resource generated by a coalition of all players. The presentinvention's enhanced “synergistic” synergistic Shapley value extendsthis concept in a novel manner and applies it in a novel manner as a wayto identify the negative impact when multiple base stations fail at thesame time. These synergistic Shapley values are proportional to both adecrease in network connectivity and a decrease in the quality ofservice, when one or more base stations fails. FIG. 4 describes how tocompute synergistic Shapley values.

Embodiments of the present invention may identify censorious macrocellsas those macrocells associated with synergistic Shapley values thatexceed a certain threshold value. The exact value of this threshold isimplementation-dependent, and may be selected by one skilled in the artor in possession of expert knowledge of an embodying application, ofresource, infrastructure, or system constraints, or of a goal or intentof the embodiment.

As mentioned above, a synergistic Shapley value is proportional to avalue of a censorious attribute of a macrocell or base station. A firstmacrocell that is censorious will, all other things equal, be associatedwith a higher synergistic Shapley value than does a second macrocellthat is not deemed censorious.

A macrocell or base station may be considered censorious when theplacement of the macrocell or base station in a cellular network is notoptimal. Reorganizing the network, to reconfigure the way that basestations or macrocells connect to each other, can significantly altersome or all macrocells' synergistic Shapley values. Embodiments of thepresent invention, may thus, by identifying censorious macrocells, helpdesigners and maintenance personnel using means known in the art toreorganize a network to reduce a total number of censorious macrocells,or to reduce synergistic Shapley values associated with certainimportant macrocells.

In one class of embodiments, synergistic Shapley values of a currentcellular network are compared to other “what-if” sets of synergisticShapley values (computed by a method of the present invention) ofalternate network topologies that connect the same base stations indifferent ways. In some cases, the what-if topologies may vary only intheir placement of macrocells known to be censorious, especiallyimportant, or most problematic.

An optimal, or most efficient, network topology may be identified as onethat comprises the lowest number of censorious macrocells or that isassociated with the lowest overall synergistic Shapley values. Suchembodiments not only identify censorious macrocells, but also identify away to reconfigure the network so as to reduce the risk posed by suchcensorious macrocells, or even so as to render then noncensorious.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, configuration data for integrated circuitry, oreither source code or object code written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Smalltalk, C++, or the like, and procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The computer readable program instructions may executeentirely on the user's computer, partly on the user's computer, as astand-alone software package, partly on the user's computer and partlyon a remote computer or entirely on the remote computer or server. Inthe latter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider). In some embodiments, electronic circuitry including,for example, programmable logic circuitry, field-programmable gatearrays (FPGA), or programmable logic arrays (PLA) may execute thecomputer readable program instructions by utilizing state information ofthe computer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

FIG. 1 shows a structure of a computer system and computer program codethat may be used to implement a method for improving robustness of acellular network in accordance with embodiments of the presentinvention. FIG. 1 refers to objects 101-115.

In FIG. 1, computer system 101 comprises a processor 103 coupled throughone or more I/O Interfaces 109 to one or more hardware data storagedevices 111 and one or more I/O devices 113 and 115.

Hardware data storage devices 111 may include, but are not limited to,magnetic tape drives, fixed or removable hard disks, optical discs,storage-equipped mobile devices, and solid-state random-access orread-only storage devices. I/O devices may comprise, but are not limitedto: input devices 113, such as keyboards, scanners, handheldtelecommunications devices, touch-sensitive displays, tablets, biometricreaders, joysticks, trackballs, or computer mice; and output devices115, which may comprise, but are not limited to printers, plotters,tablets, mobile telephones, displays, or sound-producing devices. Datastorage devices 111, input devices 113, and output devices 115 may belocated either locally or at remote sites from which they are connectedto I/O Interface 109 through a network interface.

Processor 103 may also be connected to one or more memory devices 105,which may include, but are not limited to, Dynamic RAM (DRAM), StaticRAM (SRAM), Programmable Read-Only Memory (PROM), Field-ProgrammableGate Arrays (FPGA), Secure Digital memory cards, SIM cards, or othertypes of memory devices.

At least one memory device 105 contains stored computer program code107, which is a computer program that comprises computer-executableinstructions. The stored computer program code includes a program thatimplements a method for improving robustness of a cellular network inaccordance with embodiments of the present invention, and may implementother embodiments described in this specification, including the methodsillustrated in FIGS. 1-5. The data storage devices 111 may store thecomputer program code 107. Computer program code 107 stored in thestorage devices 111 is configured to be executed by processor 103 viathe memory devices 105. Processor 103 executes the stored computerprogram code 107.

In some embodiments, rather than being stored and accessed from a harddrive, optical disc or other writeable, rewriteable, or removablehardware data-storage device 111, stored computer program code 107 maybe stored on a static, nonremovable, read-only storage medium such as aRead-Only Memory (ROM) device 105, or may be accessed by processor 103directly from such a static, nonremovable, read-only medium 105.Similarly, in some embodiments, stored computer program code 107 may bestored as computer-readable firmware 105, or may be accessed byprocessor 103 directly from such firmware 105, rather than from a moredynamic or removable hardware data-storage device 111, such as a harddrive or optical disc.

Thus the present invention discloses a process for supporting computerinfrastructure, integrating, hosting, maintaining, and deployingcomputer-readable code into the computer system 101, wherein the code incombination with the computer system 101 is capable of performing amethod for improving robustness of a cellular network.

Any of the components of the present invention could be created,integrated, hosted, maintained, deployed, managed, serviced, supported,etc. by a service provider who offers to facilitate a method forimproving robustness of a cellular network. Thus the present inventiondiscloses a process for deploying or integrating computinginfrastructure, comprising integrating computer-readable code into thecomputer system 101, wherein the code in combination with the computersystem 101 is capable of performing a method for improving robustness ofa cellular network.

One or more data storage units 111 (or one or more additional memorydevices not shown in FIG. 1) may be used as a computer-readable hardwarestorage device having a computer-readable program embodied thereinand/or having other data stored therein, wherein the computer-readableprogram comprises stored computer program code 107. Generally, acomputer program product (or, alternatively, an article of manufacture)of computer system 101 may comprise the computer-readable hardwarestorage device.

While it is understood that program code 107 for improving robustness ofa cellular network may be deployed by manually loading the program code107 directly into client, server, and proxy computers (not shown) byloading the program code 107 into a computer-readable storage medium(e.g., computer data storage device 111), program code 107 may also beautomatically or semi-automatically deployed into computer system 101 bysending program code 107 to a central server (e.g., computer system 101)or to a group of central servers. Program code 107 may then bedownloaded into client computers (not shown) that will execute programcode 107.

Alternatively, program code 107 may be sent directly to the clientcomputer via e-mail. Program code 107 may then either be detached to adirectory on the client computer or loaded into a directory on theclient computer by an e-mail option that selects a program that detachesprogram code 107 into the directory.

Another alternative is to send program code 107 directly to a directoryon the client computer hard drive. If proxy servers are configured, theprocess selects the proxy server code, determines on which computers toplace the proxy servers' code, transmits the proxy server code, and theninstalls the proxy server code on the proxy computer. Program code 107is then transmitted to the proxy server and stored on the proxy server.

In one embodiment, program code 107 for improving robustness of acellular network is integrated into a client, server and networkenvironment by providing for program code 107 to coexist with softwareapplications (not shown), operating systems (not shown) and networkoperating systems software (not shown) and then installing program code107 on the clients and servers in the environment where program code 107will function.

The first step of the aforementioned integration of code included inprogram code 107 is to identify any software on the clients and servers,including the network operating system (not shown), where program code107 will be deployed that are required by program code 107 or that workin conjunction with program code 107. This identified software includesthe network operating system, where the network operating systemcomprises software that enhances a basic operating system by addingnetworking features. Next, the software applications and version numbersare identified and compared to a list of software applications andcorrect version numbers that have been tested to work with program code107. A software application that is missing or that does not match acorrect version number is upgraded to the correct version.

A program instruction that passes parameters from program code 107 to asoftware application is checked to ensure that the instruction'sparameter list matches a parameter list required by the program code107. Conversely, a parameter passed by the software application toprogram code 107 is checked to ensure that the parameter matches aparameter required by program code 107. The client and server operatingsystems, including the network operating systems, are identified andcompared to a list of operating systems, version numbers, and networksoftware programs that have been tested to work with program code 107.An operating system, version number, or network software program thatdoes not match an entry of the list of tested operating systems andversion numbers is upgraded to the listed level on the client computersand upgraded to the listed level on the server computers.

After ensuring that the software, where program code 107 is to bedeployed, is at a correct version level that has been tested to workwith program code 107, the integration is completed by installingprogram code 107 on the clients and servers.

Embodiments of the present invention may be implemented as a methodperformed by a processor of a computer system, as a computer programproduct, as a computer system, or as a processor-performed process orservice for supporting computer infrastructure.

FIG. 2 is an undirected graph 200, representing an organization ofmacrocells of a cellular network, that may be optimized by embodimentsof the method of the present invention. FIG. 2 comprises elements200-260.

Vertices or nodes 205-230 each represent a macrocell (or a base station)of the cellular network. These macrocells are respectively labeled asnodes A-F.

Edges 235-260 each represent a bidirectional communication path betweena pair of macrocells represented respectively by nodes 205-230. In thisdocument, an edge may be labeled by the nodes it connects. For example,edge BC 240 connects node B 210 to node C 215.

The graph of FIG. 2 represents a simplified network in which each nodeis connected to only its two adjacent nodes. An equivalent graphrepresenting a real-world cellular network, however, might contain alarge number of nodes arranged in a complex web of interconnections. Insome networks, a node may be connected to more than two other nodes.

In this example, network traffic from node A 205 to node D 220 cantravel along either path ABCD (through edges 235-245), or along pathAFED (through edges 250-260). Either path has a length of 3 nodesbecause the path passes through three nodes in order to reach node D220.

If node B 210 fails, traffic between A 205 and D 220 can still travelthrough path AFED, so the length of the shortest path for such trafficdoes not change. But, if both B 210 and F 230 fail, the length of theshortest path increases (here, to an infinite value, since there is nolonger any direct or indirect path between A 205 and D 220).

Similarly, network traffic between node B 210 and F 230 can travel alongeither path BAF (through edges 235 and 260), or along path BCDEF(through edges 240-255). The shortest path BAF has a length of 2 nodes,and the longer path BCDEF has a length of 4.

If any combination of nodes C 215, D 220, or E 225 fails, trafficbetween B 210 and F 230 can still travel through the shortest BAF path,so the length of the shortest path in such case does not change. But ifnode A 205 fails, the length of the shortest path between B 210 and F230 increases because the only remaining viable path is BCDEF.

In this simple example, the censorious nature of some nodes can bemitigated by adding a new path AD (not shown in the figure) between nodeA 205 and node D 220. The result of adding such a path is that a failureby any combination of the other four nodes B 210, C 215, E 225, or F 230will no longer lengthen the shortest path between node A 205 and node D220 because the new path AD will remain the shortest possible pathbetween A 205 and D 220. But because adding a new path may consumesignificant network resources, it would generally be impossiblyresource-intensive and impossibly complex to eliminate censorious nodesin a real-world cellular network by simply connecting every node toevery other node.

Embodiments of the present invention, as described in the remainingfigures, solve this technical problem through a novel method based oncomputation of synergistic Shapley values.

FIG. 3 is a flow chart that illustrates steps of a method for improvingrobustness of a cellular network in accordance with embodiments of thepresent invention. FIG. 3 comprises steps 300-335.

In step 300, a processor of a computerized network-optimization systemreceives a representation of a topology of a cellular network. Thisrepresentation may be in a format similar to that shown in FIG. 2, inwhich a base station, macrocell, cell tower, or other type of entitycomprised by the cellular network is represented as a node or vertex ofa directed or undirected graph. For the sake of readability, theserepresented entities will be referred to here as “base stations,” butthese references should not be construed to limit embodiments of thepresent invention to only base stations.

The graph further represents a communication path between a pair of thebase stations as an edge that connects a pair of nodes that representthe pair of base stations.

Step 305 initiates an optional iterative procedure of steps 305-335. Inembodiments that comprise this iterative procedure, the procedurerepeats every time there is a change to the set of base stationscomprised by the cellular network. In other embodiments, the method ofFIG. 2 may end after a single performance of steps 310-325.

In step 310, the processor performs computations to derive a synergisticShapley value for each node of the received graph. This procedure isdescribed in greater detail in FIGS. 4 and 5.

In step 315, the processor ranks the nodes of the graph as a function oftheir corresponding synergistic Shapley values. Although the order ofthe ranking is implementation-dependent, embodiments described here rankthe nodes in decreasing order of synergistic Shapley value.

In step 320, the processor identifies censorious nodes of the graph as afunction of the synergistic Shapley values derived in step 310. Thisfunction may be implementation-dependent, using methods known in the artto infer from the synergistic Shapley values or from the ranking whichnodes are sufficiently problematic to justify an attempt to reconfigurethem.

For example, in embodiments where resource constraints limit the numberof nodes that can be reconfigured, censorious nodes may be arbitrarilyidentified as those corresponding to the highest 1% of the synergisticShapley values computed in step 310. In another case, censorious nodesmay be deemed to be those nodes corresponding to the 100 highestsynergistic Shapley values.

Other methods of selecting censorious nodes to be identified by anembodiment might include selecting nodes that are associated withsynergistic Shapley values that exceed a predetermined threshold valueor fall within a predefined range of values. In yet other cases, acensorious node might be identified as one that satisfies apredetermined combination of conditions, such as being associated with ahigher synergistic Shapley value and being physically located in ahigh-traffic geographical region or a hub city.

Many other methods are possible, and may be chosen as a function ofexpert knowledge of the cellular network or its underlying technology,or as a function of other implementation-dependent factors known tothose skilled in the art.

In step 325, the processor may forward a list of the censorious nodesidentified in step 320 to a network-management entity. Thisnetwork-management entity may be a computerized network-configuration ornetwork-maintenance entity capable of rerouting network traffic or ofintegrating the list of censorious nodes into network-configuration ornetwork-utilization analyses, reports, or proscriptive measures.

In other embodiments, the processor may forward the list of censoriousnodes to a human network administrator or other person charged withensuring the continued operation of the cellular network.

In either case, the entity receiving the list of censorious nodes may inresponse generate a report or adjust the topology of the network toreduce the number of censorious nodes or to reduce a degree of risk(which is proportional to each node's synergistic Shapley value)incurred by the existence of the censorious nodes.

In some embodiments, the method of FIG. 3 ends with step 225. But otherembodiments may continue beyond these steps, proceeding to step 330 whenthe processor receives notice of a change in the set of base stations.

In step 330, the processor receives notice of a change in the set ofbase stations. This change may comprise an addition to the set, as mightoccur when a new cell tower is brought online or when a previouslymalfunctioning or partially functioning tower returns to service. Thischange may also comprise a deletion to the set of base stations, asmight occur when an existing tower fails or becomes temporarilyunavailable while undergoing scheduled maintenance.

In step 335, the processor may update the graph that represents thecurrent topology of the cellular network. In some embodiments, theprocessor may instead receive an updated graph. In either case, theupdated graph represents the cellular network as a graph (just as theprevious graph did in step 300), and the updating comprises adding anode to or deleting a node from the graph, where the added or deletednodes represents the base station added to or deleted from the cellularnetwork in step 330.

In some cases, the updated graph may further comprise alterations to oneor more edges of the graph, representing that the network topology hasbeen revised in order to reroute data in response to the adding ordeleting of base stations. In some embodiments, rerouting may occur if,for example, a failure of a first base station completely isolates asecond base station.

In embodiments that comprise the iterative procedure of steps 305-335,the next iteration of the iterative procedure would begin with step 310.The iterative procedure would then continue as before, derivingsynergistic Shapley values for the updated network topology and thenidentifying censorious nodes to an extrinsic network-management entity.The processor would then wait until it received notification of anotherchange to the set of base stations represented by the current, updated,version of the graph.

FIG. 4 shows details of a method for computing synergistic Shapleyvalues of nodes of a graph in accordance with embodiments of the presentinvention. FIG. 4 describes step 310 of FIG. 3 in greater detail. FIG. 4comprises steps 405-430. FIG. 4 should be read in conjunction with FIG.5, which depicts an exemplary cellular network and undirected graph usedby a running example that follows the steps of FIG. 4.

FIG. 5 is an undirected graph 500, representing an organization of afive-macrocell (or five base-station) cellular network, that may beoptimized by embodiments of the method of the present invention. FIG. 5comprises elements 500-550.

FIG. 5 illustrates an exemplary five-node representation of the cellularnetwork. Elements of FIG. 5 are similar in form and function tocorresponding elements of the six-node graph of FIG. 2. Here, nodes505-525 each represent a macrocell (or a base station) of the cellularnetwork. These base stations are respectively labeled as nodes A-E.

Edges 530-550 each represent a bidirectional communication path betweena pair of macrocells represented respectively by one pair of nodes505-525. Using the naming convention of FIG. 2, edge BC 535, forexample, connects node B 510 to node C 515.

The network represented by five-node graph 500 may comprise a small,five base-station, subnetwork of a cellular network organized in a ringtopology similar to that of the six-node network of FIG. 2. The graph ofFIG. 5 contains five nodes, labeled A 505, B 510, C 515, D 520, and E525, which are connected by paths AB 530, BC 535, CD 540, DE 545, and EA550.

As described in FIG. 3, embodiments of the present invention identify“censorious” or problematic nodes of a network topology that isrepresented as a directed or undirected graph. This identification isperformed by computing novel “synergistic Shapley values” of each nodeand then ranking the nodes as a function of each node's correspondingcomputed synergistic Shapley value.

A conventional Shapley value is a concept of game theory used when agoal of a game is to fairly distribute gains and costs to a set ofactors working collaboratively. The Shapley value applies primarily insituations when the contributions of each actor are unequal. In suchcases, deriving a Shapley value for each actor provides a weighting thatensures that each actor gains at least as much as that actor would havegained by acting alone. Shapley values have been described profuselyover the years in publications related to the field of game theory andhave been applied in a variety of other fields. In the field ofeconomics, for example, a Shapley value may be characterized as anexpectation of a marginal value or contribution. In general, a Shapleyvalue is expressed as a fraction.

The present invention comprises a novel application of the concept ofsynergistic Shapley values in order to identify how much disruption to anetwork would occur when a particular node of that network becomesunavailable. In embodiments of the present invention, the Shapleycomputation is extended to account for the effect of scenarios in whichmultiple nodes become unavailable at the same time. These enhanced,“synergistic,” synergistic Shapley values may be computed by the methodof FIG. 4.

A common expression of a function ϕi that returns a Shapley value of aplayer I is:

$\begin{matrix}{{\phi_{i}(v)} = {\sum\limits_{C \subseteq {N - i}}{\frac{{{C}!}{\left( {n - {C} - 1} \right)!}}{n!}\left\{ {{v\left( {C\bigcup\left\{ i \right\}} \right)} - {v(C)}} \right\}}}} & (1)\end{matrix}$

Here:

-   -   N is a set of players,    -   n is a total number of players in N,    -   i is a player comprised by N,    -   C is a subset of N that consists of players working in        collaboration, and    -   |C| is the number of collaborating players in a subset C.

A conventional game-theory application of Eq. (1) may comprise using theequation to compute a Shapley value for a player i as a sum taken overall subsets N−i (all subsets of N that do not include player i). As isknown in the field of mathematics, the number of distinct subsets C(N−i) is equal to the number of combinations of n objects taken |C| at atime. Function v(C), as it is known in the art, returns a “coalitionworth” value of subset C.

Embodiments of the present invention apply an analogous computation toproduce a synergistic Shapley value of a node i that may be ascribedmeaning within the context of the present invention. For example,embodiments described below interpret v(C) in a novel manner in order toadd “synergistic” weighting to derived synergistic Shapley values (thatis, to adjust a synergistic Shapley value of a cellular network toaccount for concurrent multiple base-station failures).

For example, in the six-node graph of FIG. 2, an embodiment of thepresent invention might be used to derive synergistic Shapley values forone node i. Here, the set of all players N would the set of all nodes(A, B, C, D, E, F), n would equal 6 (the number of nodes in N), and eachsubset C would be a subset of N that does not include node i.

In this example, if Eq. (1) is used to derive a synergistic Shapleyvalue for i=node A 205, the equation might produce terms for an emptysubset (|C|=0), for each of the five subsets that contain one node(|C|=1) but do not contain node A 205, and so forth, through a final,sixth, pass that processes the single five-node subset (C=5) that doesnot contain node A 205.

In embodiments of the present invention, the function v(C) returns avalue for each subset C of nodes. This value is equal to the number ofpairs of nodes in N for which the shortest path between each pair ofnodes increases when all the nodes of subset C become nonfunctional.

For example, if subset C={BD}, v(C) returns the number of base-stationpairs that would be forced to exchange data through a longer path shouldnode B 210 and node D 220 both become inoperable. Here, the shortestpath between node A 205 and node C 225 lengthens (that is, the shortestpath between those two nodes becomes impossible). Due to similarreasons, the shortest path between node C 215 and E 225 lengthens andthe shortest path between node C 215 and node F 230 also lengthens.Neither the shortest path between nodes A 205 and D 220, nor theshortest path between nodes D 220 and E 225 lengthens. Because theshortest paths between three pairs of nodes lengthen when the nodes insubset {BD} simultaneously fail, v(C)=v({BD})=3. Further examples offunction v( ) are presented in greater detail below, in the runningexample based on FIG. 5.

A synergistic Shapley value for any node i of set N, as computed by Eq.(1), is thus proportional to: (i) a decrease in connectivity of thenetwork N when the base station corresponding to node i becomesnonoperational; and (ii) a decrease in the network's overall quality ofservice when the base station corresponding to node i becomesnonoperational.

Embodiments of the present invention therefore apply the Shapleyfunction in a novel way by deeming base stations of a cellular networkto be collaborating players of a game governed by the theorems of gametheory, and in this way use the Shapley function to produce values thatare proportional to how potentially problematic (or censorious) is eachbase station of the network.

In step 405, the processor, having received a graph that represents atopology of all or part of the cellular network, enumerates all possiblesubsets of nodes of the received graph.

In the running example where the received graph is the five-node graph500 of FIG. 5, this enumeration might be organized into five groups ofsubsets (grouped solely for pedagogical reasons as a function of thenumber of elements |C| in each subset C):

-   -   (|C|=0): { }    -   (|C|=1): {A} {B} {C} {D} {E}    -   (|C|=2): {AB} {AC} {AD} {AE} {BC} {BD} {BE} {CD} {CE} {DE}    -   (|C|=3): {ABC} {ABD} {ABE} {ACD} {ACE} {ADE} {BCD} {BCE} {BDE}        {CDE}    -   (|C|=4) {ABCD} {ABCE} {ABDE} {ACDE} {BCDE} and    -   (|C|=5) {ABCDE}.

One way to conceptualize this enumeration is as a list of combinationsof n objects taken |C| at a time, where n is the total number of nodes(5) comprised by the set of all nodes N, and |C| is the number of nodesin any subset C⊂N.

Step 410 begins an iterative procedure of steps 410-430. This iterativeprocedure is performed once for each node in the received graph. In therunning example that processes the topology of FIG. 5, this iterativeprocedure would be performed five times, once each for nodes A 505, B510, C 515, D 520, and E 525.

The description of FIG. 4 presented in this document will, forillustrative purposes, walk through this iterative procedure once,showing how to derive a synergistic Shapley value for node B 510. In afull implementation, four other iterations of the iterative procedurewould each perform analogous operations on each of the other four nodesin order to compute synergistic Shapley values for those other fournodes. Steps 315-320 of FIG. 3 would then rank and compare the fivevalues in order to identify censorious nodes.

In step 415, the processor begins a sequence of steps that evaluate Eq.(1) in order to derive a synergistic Shapley value for node i. In therunning example, node i, during the current iteration of the iterativeprocedure of steps 410-430, is node B 210 of the graph of FIG. 5.

This sequence of steps begins with the processor identifying theshortest path between each pair of nodes of the set N of all nodes. Inour running example, N consists of all nodes represented by the graph ofFIG. 5: {A, B, C, D, E}. These computations may be performed by anymeans known in the art, such as by an algorithm that identifies shortestpath lengths by counting nodes as it traverses a graph.

In one example, the shortest path between nodes A 505 and B 510 isdeemed to have a length of 1 because the shortest path traverses onlyone node. Similarly, the shortest path between nodes A 505 and C 515 hasa length of 2 because the first link on the shortest path connects nodeA 505 to node B 510 and the second link on the shortest path connectsnode B 510 to node C 515. And the shortest path between node B 510 andnode D 520 has a length of 2 because the shortest path between those twonodes must pass through node C 515 and node D 520. The shortest lengthsof all other subsets C are computed in the same manner.

In some embodiments, step 415 may be performed at a different point in amethod of FIG. 3 or FIG. 4, or may be combined with another step, suchas step 420. In some embodiments, this step 415 may be performed only tofacilitate a performance of step 420, or only to enumerate all possiblesubsets C. The enumeration of step 415 is presented here as a distinctstep, performed at a distinct point in a method of FIG. 4, solely inorder to facilitate a clear explanation of the steps required to derivesynergistic Shapley values in accordance with embodiments of the presentinvention.

In step 420, the processor computes v(C) for each subset C enumerated instep 400. As described above, function v(C) returns a value for eachsubset C of the set of nodes N. As interpreted by embodiments of thepresent invention, v(C) returns the number of node pairs of N for whichthe shortest path increases in length when all the nodes in C areremoved from the graph. If, for example, the shortest paths betweenthree pairs of nodes increase in length when all the nodes of C areremoved from the graph, then v(C)=3.

For example, a failure of node B 510 of FIG. 5 blocks traffic throughnode B 510, making impossible the previously shortest path from A 505 toC 515 through B 510. A new shortest path from A 505 to C 515 must nowpass through nodes D 520 and E 525, resulting in a lengthening of theshortest path between node A 505 and C 515. The function v({B}) thusreturns a value of 1, the number of shortest paths between other pairsof nodes that lengthens.

In the running example based on the five-node graph of FIG. 5, theprocessor enumerated 32 distinct subsets C⊂N in step 400. Computing v(C)for these subsets yields the following values of v( ):

v({ }) = 0 v({A, B, C, D, E}) = 0 v({A}) = 1 v({B}) = 1 v({C}) = 1v({D}) = 1 v({E}) = 1 v({A, B}) = 0 v({A, C}) = 2 v({A, D}) = 2 v({A,E}) = 0 v({B, C}) = 0 v({B, D}) = 2 v({B, E}) = 2 v({C, D}) = 0 v({C,E}) = 2 v({D, E}) = 0 v({A, B, C}) = 0 v({A, B, D}) = 1 v({A, B, E}) = 0v({A, C, D}) = 1 v({A, C, E}) = 1 v({A, D, E}) = 0 v({B, C, D}) = 0v({B, C, E}) = 1 v({B, D, E}) = 1 v({C, D, E}) = 0 v({A, B, C, D}) = 0v({A, B, C, E}) = 0 v({A, B, D, E}) = 0 v({A, C, D, E}) = 0 v({B, C, D,E}) = 0

In step 425, the processor, by inserting the previously derived valuesinto Eq. (1), determines a term of a synergistic Shapley value for nodeI (node B 510), where each of these terms corresponds to a subset C−i ofset N={A, B, C, D, E}.

In our running example, subset C may thus be any of the following 16subsets, none of which include node B 510: { }, {A}, {C}, {D}, {E},{A,C}, {A,D}, {A,E}, {C,D}, {C,E}, {D,E}, {A,C,D}, {A,C,E}, {A,D,E},{C,D,E}, {A,C,D,E}. As above, these subsets are organized, forpedagogical purposes, into groups that each contain all subsets thathave a same number of elements.

Plugging values of n=5, i=node B 210, and N={A,B,C,D,E} into Eq. (1)yields the following 16 terms of a synergistic Shapley value of node B210, each of which corresponds to one subset C−i and each of whichrepresents one term over which is summed Eq. (1):

-   -   |C|=0:        -   C={ }(empty set): [0! (5−0−1)!/5!]*[v({B})−v({            }]=[4!/5!]*[1−0]=24/120    -   |C|=1:        -   C={A}: [1! (5−1−1)!/5!] *[v({A,B})−v({A})]=[3!/5!] *[0−0]=0        -   C={C}: [1! (5−1−1)!/5!] *[v({C,B})−v({C})]=[3!/5!]            *[0−1]=6/120 (−1)        -   C={D}: [1! (5−1−1)!/5!] *[v({D,B})−v({D})]=[3!/5!]            *[2−1]=(6/120)*1=6/120        -   C={E}: [1! (5−1−1)!/5!] *[v({E,B})−v({E})]=[3!/5!]            *[2−1]=(6/120)*1=6/120    -   |C|=2:        -   C={A,C}: [2! (5−2−1)!/5!] *[v({A,B,C})−v({A,C})]=[4/5!]            *[0−2]=4/120*(−2)        -   C={A,D}: [2! (5−2−1)!/5!] *[v({A,B,D})−v({A,D})]=[4/5!]            *[1−2]=4/120*(−1)        -   C={A,E}: [2! (5−2−1)!/5!] *[v({A,B,E})−v({A,E})]=[4/5!]            *[0−0]=4/120*(0)        -   C={C,D}: [2! (5−2−1)!/5!] *[v({B,C,D})−v({C,D})]=[4/5!]            *[0−0]=4/120*(0)        -   C={C,E}: [2! (5−2−1)!/5!] *[v({B,C,E})−v({C,E})]=[4/5!]            *[1−2]=4/120*(−1)        -   C={D,E}: [2! (5−2−1)!/5!] *[v({B,D,E})−v({D,E})]=[4/5!]            *[1−0]=4/120*(1)    -   |C|=3:        -   C={A,C,D}: [3! (5−3−1)!/5!]            *[v({A,B,C,D})−v({A,C,D})]=[3!/5!] *[0−1]=6/120*(−1)        -   C={A,C,E}: [3! (5−3−1)!/5!]            *[v({A,B,C,E})−v({A,C,E})]=[3!/5!] *[0−1]=6/120*(−1)        -   C={A,D,E}: [3! (5−3−1)!/5!]            *[v({A,B,D,E})−v({A,D,E})]=[3!/5!] *[0−0]=6/120*(0)        -   C=(C,D,E): [3! (5−3−1)!/5!]            *[v({B,C,D,E})−v({C,D,E})]=[3!/5!] *[0−0]=6/120*(0)    -   |C|=4:        -   C={A,C,D,E}: [4! (5−4−1)!/5!]            *[v({A,B,C,D,E})−v({A,C,D,E})]=[24/120] *[0−0]=0

The final synergistic Shapley value for node B 510 may thus be computedby summing the individual values derived above in accordance with Eq.(1). This produces the value:

-   -   24/120+(0−6/120+6/120+6/120)+(−8/120−4/120+0+0+−4/120+4/120)+(−6/120−6/120+0+0)+0=24/120+6/120−12/120−12/120+0=6/120=1/20

The synergistic Shapley value for node B 210 is thus 1/20.

In our running example, four other iterations of the iterative procedureof steps 410-430 return synergistic Shapley values for nodes A, C, D,and E in the same manner. The method of FIG. 4 then ends and embodimentsof the present invention continue with step 315 of FIG. 3.

What is claimed is:
 1. A network-optimization system comprising aprocessor, a memory coupled to the processor, and a computer-readablehardware storage device coupled to the processor, the storage devicecontaining program code configured to be run by the processor via thememory to implement a method for improving robustness of a cellularnetwork, the method comprising: receiving an undirected graph thatrepresents a topology of a cellular network, where the graph representsa pair of base stations of the network as a pair of nodes, and where thegraph represents a communications path between the pair of base stationsas an edge connecting the pair of nodes; deriving, for each node of thegraph, a corresponding Shapley value of a plurality of Shapley values;identifying a first node of the graph as a censorious node if a Shapleyvalue that corresponds to the first node exceeds a predeterminedthreshold value; updating the graph by rearranging one or morecensorious nodes of the graph so as to reduce the Shapley values thatcorrespond to the one or more censorious nodes; and directing anetwork-management entity, in response to the updating, to revise thecellular network such that the topology of the updated network matchesthe topology of the updated graph.
 2. The system of claim 1, furthercomprising: repeating the deriving on two or more permutations of theundirected graph, where each permuted graph differs from the undirectedgraph only in the placement of macrocells previously identified as beingcensorious; identifying, as a function of the repeating, an updated setof censorious nodes for each permutation; selecting, from the two ormore permutations, a least-censorious permutation, where theleast-censorious permutation comprises fewer censorious nodes than anyother permutation; replacing the revised topology with a permutedtopology that corresponds to the least-censorious permutation; anddirecting the network-management entity to revise the cellular networksuch that the topology of the updated network matches the permutedtopology.
 3. The system of claim 1, further comprising: identifying thata failed base station of the cellular network is not operating; deletingfrom the graph a failed node that represents the failed base station;deleting all edges of the graph that connect the failed node to othernodes of the graph; and repeating the deriving, identifying, updating,and revising.
 4. The system of claim 1, further comprising: identifyingthat a new base station has been added to the cellular network, wherethe new base station communicates with the cellular network through oneor more new communications paths; adding to the graph a new node hatrepresents the new base station; adding to the graph a new edge thatconnects the new node to an existing node of the graph, where the newedge represents a communication path between the new base station and anexisting base station of the cellular network; and repeating thederiving, identifying, updating, and revising.
 5. The system of claim 1,where the deriving a Shapley value Ø_(i) that corresponds to a node i ofthe graph comprises evaluating an equation of the form:$\varnothing_{i} = {\sum\limits_{C \in {N - i}}{\frac{{{C}!}*{\left( {n - {C} - 1} \right)!}}{n!}*\left\{ {{v\left( {C\bigcup\left\{ i \right\}} \right)} - {v(C)}} \right\}}}$where N is a set of all nodes of the graph, n is a number of nodescontained in set N, C is a subset of set N, |C| is a number of nodescontained in subset C, N−i is a subset of set N that does not includenode i, and v(C) is a function that returns a value proportional to anamount of disruption to the cellular network resulting from a failure ofall base stations of the cellular network that are represented by nodesof subset C.
 6. The system of claim 5, where the value by function v(C)indicates how many base-station pairs of the cellular network are forcedto exchange data through a longer path as a result of a failure of allbase stations of the cellular network that are represented by nodes ofsubset C.
 7. A method for improving robustness of a cellular network,the method comprising: receiving an undirected graph that represents atopology of a cellular network, where the graph represents a pair ofbase stations of the network as a pair of nodes, and where the graphrepresents a communications path between the pair of base stations as anedge connecting the pair of nodes; deriving, for each node of the graph,a corresponding Shapley value of a plurality of Shapley values;identifying a first node of the graph as a censorious node if a Shapleyvalue that corresponds to the first node exceeds a predeterminedthreshold value; and updating the graph by rearranging one or morecensorious nodes of the graph so as to reduce the Shapley values thatcorrespond to the one or more censorious nodes; and directing anetwork-management entity, in response to the updating, to revise thecellular network such that the topology of the updated network matchesthe topology of the updated graph.
 8. The method of claim 7, furthercomprising: repeating the deriving on two or more permutations of theundirected graph, where each permuted graph differs from the undirectedgraph only in the placement of macrocells previously identified as beingcensorious; identifying, as a function of the repeating, an updated setof censorious nodes for each permutation; selecting, from the two ormore permutations, a least-censorious permutation, where theleast-censorious permutation comprises fewer censorious nodes than anyother permutation; replacing the revised topology with a permutedtopology that corresponds to the least-censorious permutation; anddirecting the network-management entity to revise the cellular networksuch that the topology of the updated network matches the permutedtopology.
 9. The method of claim 7, further comprising: identifying thata failed base station of the cellular network is not operating; deletingfrom the graph a failed node that represents the failed base station;deleting all edges of the graph that connect the failed node to othernodes of the graph; and repeating the deriving, identifying, updating,and revising.
 10. The method of claim 7, further comprising: identifyingthat a new base station has been added to the cellular network, wherethe new base station communicates with the cellular network through oneor more new communications paths; adding to the graph a new node thatrepresents the new base station; adding to the graph a new edge thatconnects the new node to an existing node of the graph, where the newedge represents a communication path between the new base station and anexisting base station of the cellular network; and repeating thederiving, identifying, updating, and revising.
 11. The method of claim7, where the deriving a Shapley value Ø_(i) that corresponds to a node iof the graph comprises evaluating an equation of the form:$\varnothing_{i} = {\sum\limits_{C \in {N - i}}{\frac{{{C}!}*{\left( {n - {C} - 1} \right)!}}{n!}*\left\{ {{v\left( {C\bigcup\left\{ i \right\}} \right)} - {v(C)}} \right\}}}$where N is a set of all nodes of the graph, n is a number of nodescontained in set N, C is a subset of set N, |C| is a number of nodescontained in subset C, N−i is a subset of set N that does not includenode i, and v(C) is a function that returns a value proportional to anamount of disruption to the cellular network resulting from a failure ofall base stations of the cellular network that are represented by nodesof subset C.
 12. The method of claim 11, where the value returned byfunction v(C) indicates how many base-station pairs of the cellularnetwork are forced to exchange data through a longer path as a result ofa failure of all base stations of the cellular network that arerepresented by nodes of subset C.
 13. The method of claim 7, furthercomprising: ordering the Shapely values derived for all nodes of thegraph, where the ordering ranks the Shapely values corresponding to allnodes of the graph in decreasing order; and identifying as censoriousnodes a subset of nodes of the graph that correspond to a predeterminednumber of the highest-ranked Shapely values.
 14. The method of claim 7,further comprising providing at least one support service for at leastone of creating, integrating, hosting, maintaining, and deployingcomputer-readable program code in the computer system, where thecomputer-readable program code in combination with the computer systemis configured to implement the receiving, the deriving, the identifying,the updating, and the revising.
 15. A computer program product,comprising a computer-readable hardware storage device having acomputer-readable program code stored therein, the program codeconfigured to be executed by a network-optimization system comprising aprocessor, a memory coupled to the processor, and a computer-readablehardware storage device coupled to the processor, the storage devicecontaining program code configured to be run by the processor via thememory to implement a method for improving robustness of a cellularnetwork, the method comprising: receiving an undirected graph thatrepresents a topology of a cellular network, where the graph representsa pair of base stations of the network as a pair of nodes, and where thegraph represents a communications path between the pair of base stationsas an edge connecting the pair of nodes; deriving, for each node of thegraph, a corresponding Shapley value of a plurality of Shapley values;identifying a first node of the graph as a censorious node if a Shapleyvalue that corresponds to the first node exceeds a predeterminedthreshold value; and updating the graph by rearranging one or morecensorious nodes of the graph so as to reduce the Shapley values thatcorrespond to the one or more censorious nodes; and directing anetwork-management entity, in response to the updating, to revise thecellular network such that the topology of the updated network matchesthe topology of the updated graph.
 16. The computer program product ofclaim 15, further comprising: repeating the deriving on two or morepermutations of the undirected graph, where each permuted graph differsfrom the undirected graph only in the placement of macrocells previouslyidentified as being censorious; identifying, as a function of therepeating, an updated set of censorious nodes for each permutation;selecting, from the two or more permutations, a least-censoriouspermutation, where the least-censorious permutation comprises fewercensorious nodes than any other permutation; replacing the revisedtopology with a permuted topology that corresponds to theleast-censorious permutation; and directing the network-managemententity to revise the cellular network such that the topology of theupdated network matches the permuted topology.
 17. The computer programproduct of claim 15, further comprising: identifying that a failed basestation of the cellular network is not operating; deleting from thegraph a failed node that represents the failed base station; deletingall edges of the graph that connect the failed node to other nodes ofthe graph; and repeating the deriving, identifying, updating, andrevising.
 18. The computer program product of claim 15, furthercomprising: identifying that a new base station has been added to thecellular network, where the new base station communicates with thecellular network through one or more new communications paths; adding tothe graph a new node that represents the new base station; adding to thegraph a new edge that connects the new node to an existing node of thegraph, where the new edge represents a communication path between thenew base station and an existing base station of the cellular network;and repeating the deriving, identifying, updating, and revising.
 19. Thecomputer program product of claim 15, where the deriving a Shapley valueØ_(i) that corresponds to a node i of the graph comprises evaluating anequation of the form:$\varnothing_{i} = {\sum\limits_{C \in {N - i}}{\frac{{{C}!}*{\left( {n - {C} - 1} \right)!}}{n!}*\left\{ {{v\left( {C\bigcup\left\{ i \right\}} \right)} - {v(C)}} \right\}}}$where N is a set of all nodes of the graph, n is a number of nodescontained in set N, C is a subset of set N, |C| is a number of nodescontained in subset C, N−i is a subset of set N that does not includenode i, and v(C) is a function that returns a value proportional to anamount of disruption to the cellular network resulting from a failure ofall base stations of the cellular network that are represented by nodesof subset C.
 20. The computer program product of claim 19, where thevalue returned by function v(C) indicates how many base-station pairs ofthe cellular network are forced to exchange data through a longer pathas a result of a failure of all base stations of the cellular networkthat are represented by nodes of subset C.